[tex](y-\sec x)\,\mathrm dx+\tan x\,\mathrm dy=0[/tex]
Divide both side by [tex]\mathrm dx[/tex] and rearrange terms to get a linear ODE;
[tex]\tan x\dfrac{\mathrm dy}{\mathrm dx}+y=\sec x[/tex]
Multiply both sides by [tex]\cos x[/tex]:
[tex]\sin x\dfrac{\mathrm dy}{\mathrm dx}+\cos x\,y=1[/tex]
The left side can be condensed as the derivative of a product:
[tex]\dfrac{\mathrm d}{\mathrm dx}(\sin x\,y)=1[/tex]
Integrate both sides, then solve for [tex]y[/tex]:
[tex]\sin x\,y=x+C\implies\boxed{y(x)=x\csc x+C\csc x}[/tex]
